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9.21: The Entropy Change for A Spontaneous Process at Constant E and V

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    For any spontaneous process, we have \(dE={dq}^{spon}\)+\({dw}^{spon}\), which we can rearrange to \({dq}^{spon}=dE-{dw}^{spon}\). Substituting our result from Section 9.15, we have

    \[\hat{T}dS>dE-dw^{spon} \nonumber \] (spontaneous process)

    If the energy of the system is constant throughout the process, we have \(dE=0\) and

    \[\hat{T}{\left(dS\right)}_E>-dw^{spon} \nonumber \] (spontaneous process, constant energy)

    The spontaneous work is the sum of the pressure–volume work and the non-pressure–volume work, \(\ dw^{spon}={dw}^{spon}_{PV}+{dw}^{spon}_{NPV}\). If we introduce the further condition that the spontaneous process occurs while the volume of the system remains constant, we have \({dw}^{spon}_{PV}=0\). Making this substitution and repeating our earlier result for a reversible process, we have the parallel relationships

    \[{\left(dS\right)}_{EV}>\frac{-dw^{spon}_{NPV}}{\hat{T}} \nonumber \] (spontaneous process, constant \(E\) and \(V\))

    \[{\left(dS\right)}_{EV}=\frac{-dw^{spon}_{NPV}}{\hat{T}} \nonumber \] (reversible process, constant \(E\) and \(V\))

    (For a reversible process, \(T=\hat{T}\).) If the spontaneous process occurs while \(\hat{T}\) is constant, summing the incremental contributions to a finite change of state produces the parallel relationships

    \[{\left(\Delta S\right)}_{EV}>\frac{-w^{spon}_{NPV}}{\hat{T}} \nonumber \] (spontaneous process, constant \(E\), \(V\), and \(\hat{T}\))

    \[{\left(\Delta S\right)}_{EV}=\frac{-w^{spon}_{NPV}}{\hat{T}} \nonumber \] (reversible process, constant \(E\), \(V\), and \(\hat{T}\))

    Constant \(\hat{T}\) corresponds to the common situation in chemical experimentation in which we place a reaction vessel in a constant-temperature bath. If we introduce the further condition that only pressure–volume work is possible, we have \(dw^{spon}_{NPV}=0\). The parallel relationships become

    \[{\left(dS\right)}_{EV}>0 \nonumber \] (spontaneous process, constant \(E\) and \(V\), only \(PV\) work)

    \[{\left(dS\right)}_{EV}=0 \nonumber \] (reversible process, constant \(E\) and \(V\), only\(\ PV\) work)

    If the energy and volume are constant for a system in which only pressure–volume work is possible, the system is isolated. The conditions we have just derived are entirely equivalent to our earlier conclusions that \(dS=0\) and \(dS>0\) for an isolated system that is at equilibrium or undergoing a spontaneous change, respectively. Summing the incremental contributions to a finite change of state produces the parallel relationships

    \[{\left(\Delta S\right)}_{EV}>0 \nonumber \] (spontaneous process, only \(PV\) work)

    \[{\left(\Delta S\right)}_{EV}=0 \nonumber \] (reversible process, only \(PV\) work)

    The validity of these expressions is independent of any variation in either \(T\) or \(\hat{T}\).

    This page titled 9.21: The Entropy Change for A Spontaneous Process at Constant E and V is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform.